Iterative Determination of Spar Lines Static Equilibrium and Improved Dynamic Modeling by Fractional Derivatives. Georgios I.


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1 RICE UNIVERSITY Iterative Determination of Spar Lines Static Equilibrium and Improved Dynamic Modeling by Fractional Derivatives By Georgios I. Evangelatos A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE MASTER OF SCIENCE APPROVED, THESIS COMMITTEE: Dr. Sparjosfol, D. Chairman Professor of Civil and Mechanical Engineering Dr. Dick Andrew, J Assistant Professor of Mechanical Engineering Qpfev^^f Dr. Padgett Jamie, E Assistant Professor of Civil and Environmental Engineering Houston, Texas USA May 2009
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3 Abstract Iterative Determination of Spar Lines Static Equilibrium and Improved Dynamic Modeling by Fractional Derivatives By Georgios I. Evangelatos Polyester mooring systems are used as permanent systems offloatingproduction systems and offshore structures. Compared to other mooring systems, polyester has a highly non linear behavior, thus complicating the overall design. Three important parameters affect the polyester rope stiffness; the mean load, the load range, and the frequency of the loading. Ignoring the 'true' stiffness as influenced by these parameters, may lead to underestimate the load induced riser stresses and damage. Procedures for determining mooring line stiffness that is representative of the in service conditions are developed herein. Two stiffness values, the 'static' and 'dynamic', are iteratively calculated from real time data, and are correlated with laboratory tests. Furthermore, note that fractional derivative models have been extensively proposed in literature for accurately capturing frequency dependent behavior of materials. In this context, a modified Newmark algorithm that takes advantage of the GrunwaldLetnikov fractional derivative representation is developed to treat related structural dynamic problems. In addition, a statistical linearization approach is also developed for random vibration treatment of such systems. The modified Newmark Algorithm is used to conduct Monte Carlo studies demonstrating the reliability of the statistical linearization solution. 2
4 Acknowledgments I would like to thank Professor Pol D. Spanos for his patience, skillful guidance and consistent support in these last two years, but above all I would like to thank him for his incredible mentorship, the environment of success that he promotes, and the chance he gave me to collaborate with him. Thanks also are due to Dr Jamie E. Padgett and Dr Andrew J. Dick for serving on the thesis committee and providing me constructive comments. I also want to thank all my friends and professors at Rice University for making my stay here an incredible, once in a life time, experience. Special thanks go to my brothers Panagiotis and Dionisis for their never ending love and support and to my parents Ioannis Evangelatos and Virginia Evangelatos for their struggles to give me a good life. 3
5 Table of Contents CHAPTER 1 Introduction General Remarks, Statement of the problem Organization of the Thesis 12 CHAPTER 2 Polyester Mooring Systems Background 13 CHAPTER 3 Modeling of the Polyester Lines Introduction Modeling of the polyester mooring system problem Mathematical Background GaussNewton non linear minimization algorithm 33 CHAPTER 4 Numerical Results Pertaining to Hurricane Katrina Modeling connectors as catenaries Numerical Results Pertaining to HurricaneKatrina Data Numerical Results on the 'Dynamic' stiffness Signal Processing of the Results 47 CHAPTER 5 Non Linear Dynamic Systems with Damping Forces Governed by Fractional Derivatives Preliminary Remarks 51 4
6 5.2 Literature Survey on Fractional Derivatives Mathematical representation of fractional derivatives Modified Newmark Algorithm for integration of equations in time Numerical Results 62 CHAPTER 6 Deterministic and Statistical Linearization Deterministic Linearization Statistical Linearization Numerical Results for random excitation. 71 CHAPTER 7 The Improved Model via Fractional Derivatives 72 CHAPTER 8 Concluding Remarks 83 References 5
7 List of Figures 3.1. The line from the fairlead point of the platform to the anchoring point A typical polyester mooring line in a inplane depiction The infinitesimal submerged rope section taking of from the sea floor Displacements and rotations with respect to a certain axis system Iterative procedure of 'shooting down the line' Infinitesimal submerged rope segment under gravitational force < Maximum fairlead tension in Kips versus the number of 20min files from the 25 th morning to 28th Surge fluctuation and mean value over a 20min period of time Sway fluctuation and mean value over a 20min period of time Heave fluctuation and mean value over a 20min period of time Effective 'elastic' length versus 'static' stiffness of polyester rope 'Static' stiffness fluctuation and mean value over a 20min period of time Least squares approximation through a 'quasi static' analysis 'Dynamic' stiffness fluctuation over a 20min period of time Spectral estimation via PWelch matlab function of signal at figure 'Dynamic' stiffness fluctuation over a 20min period of time Spectral estimation via PWelch matlab function of signal at figure Filtered signal of figure 13 with 5 th order Butterworth IIR filter for low, band and high pass frequencies Filtered signal of figure 15 with 5* order Butterworth IIR filter for low, band and high pass frequencies Displacement of the Duffing Oscillator under sinusoidal load 62 6
8 5.2. Damping force versus time calculated by the product cd a x(t) The GL values of equation (5.13) calculated recursively using equation (5.20) Amplitude response for a single degree of freedom system with different values of fractional derivatives Amplitude response of a single degree of freedom system with different fractional derivatives Monte Carlo points using equation (5.27) with 3 past terms corresponding to order 0 vis a vis statistical linearization results Monte Carlo points using equation (5.27) with 8 past terms corresponding to order 10' 2 vis A vis statistical linearization results Monte Carlo points using equation (5.27) with 11 past terms corresponding to order 10' 2 Vis a vis statistical linearization results Monte Carlo points using equation (5.27) with 12 past terms corresponding to order 10" 2 Vis a vis statistical linearization results Monte Carlo points using equation (5.27) with 13 past terms corresponding order io' 2 vis a vis statistical linearization results Monte Carlo points using equation (5.27) with 14 past terms corresponding to order 10" 3 vis a vis statistical linearization results 'Dynamic' stiffness corresponding to the fractional derivative term 'Dynamic' stiffness corresponding to the fractional derivative term 81 7
9 List of Tables 4.1. Error results from commercial F.E. package and the custom program for small connecting segments Error results from commercial F.E. package and the custom program for connecting segments of reasonable length Error results for commercial F.E. package and the custom program for large connecting segments Error results for commercial package and the custom program for quite large connecting segments Error results for commercial F.E. package and custom program for in service line at the gulf of Mexico with largely modeled connecting segments Error results for commercial F.E. package and custom algorithm for one segment line from the sea floor to the platform 40 8
10 CHAPTER 1: Introduction 1.1 General Remarks Polyester mooring systems were first used in the USA in 2004 as a system for anchoring Floating Production Systems (FPS) in the gulf of Mexico. Steel chain, and wire or spiral strand are commonly used as permanent mooring systems in offshore structures. However, the fact that steel is quite heavy even if submerged, poses serious challenges to the design of the hull. The hull and the jacks in the offshore structure must be designed in such a manner to withstand up to 2000 tons of additional vertical force. Therefore, the weight and the cost to build and to move on site such structures is quite high. Polyester mooring systems on the other hand are buoyant when submerged and provide the same, if not, more breaking strength as the steel lines. Furthermore, the use of them results into smaller platforms and ground chains, smaller chain jacks and fairleads, which also produces cost savings beyond the impact on the hull. The deployment of polyester mooring systems is quite easier than the steel wires, and therefore its usage is highly preferred in ultra deep water explorations (beyond 1100m). The design of a polyester mooring line however, is significantly different than that of a steel wire or spiral strand. First, the polyester rope stiffness is a function of many variables unlike steel. Further, for a Floating Production System (FPS), currents of the functioning environment will usually control the design. Unlike wind and wave loads, which induce mean and dynamic loading on the structure, current loads tend to produce a large mean load on the hull that is inline with the current flow, plus the transverse hull motions due to vortex induced motions. Thus, the static stiffness of the polyester rope becomes much more important than the dynamic stiffness. To date, the industrial community has collected most of the test data 9
11 on dynamic rope stiffness and thus little is known about the static stiffness. Steel chain, wire or spiral strand produce a nearly linear loadextension curve over the range of loading involved in most mooring designs. A polyester mooring tether, however, has a highly nonlinear loadextension curve, thus complicating the mooring line and riser design. To obtain stiffness properties for the initial stages of the design, data can be solicited from various rope manufacturers, and a literature survey can be conducted. The majority of the data that exist though, focus on dynamic axial stiffness. Little, information can be found on static stiffness of polyester ropes. There are some references in texts but often the reported rate of loading is comparably short in terms of what is expected in the field, or the mean load or rate of loading is not quantified, and therefore definitive conclusions can not be drawn. 1.2 Statement of the problem Due to the fact that the platform is subjected to wind and wave loadings that tend to produce dynamic affects on the platform, but also is subjected to currents that tend to cause static deformations, the twomodulus method has been applied by the industrial design teams. The use of a higher storm modulus for checking line tensions and a lower drift modulus for checking vessel offsets. Although this may be adequate for the mooring design, it can be inadequate for riser design in extreme events or everyday fatigue sea states. Furthermore, since the restoring force for a polyester taut line system is primarily derived from the extension in the line, it is important to adequately understand how the stiffness of the tether will vary over all expected conditions. Three main parameters appear to have an affect on the polyester rope stiffness; the mean load, the load range in which the line is subjected and the frequency in which the load is applied. Ignoring the 10
12 true value of the stiffness as influenced by these parameters, and using a simple bounding method may estimate incorrectly the resulting riser stress and fatigue life. Thus, understanding the rope stiffness is important for proper mooring and riser system design. The dynamic stiffness is different from the static stiffness, and since the mooring system is controlled by current conditions, the static stiffness should be accurately known. In this thesis an approach is described for determining the static and dynamic stiffness of polyester lines in anchoring systems of offshore structures. Assessing the stiffness of a polyester rope in field conditions poses non trivial challenges. Specifically, materials of this kind have large breaking strength, and are almost buoyant when submerged. Furthermore, the stiffness depends on the excitation and the mean load of it. Due to the fact that these materials are non linear, a scanning of frequencies in the laboratory to determine the stiffness is not feasible. In this context, a deterministic iterative approach involving appropriate catenary equations is used along with field data from an offshore structure to derive a reliable estimate of the effective static and dynamic stiffness of polyester mooring lines. Furthermore, the fact that hysterisis and frequency dependence are accurately captured by the use of fractional derivative, such a model is exploited. It is shown that the use of the GrunwaldLetnikov definition of fractional derivative in conjunction with Newmark numerical integration scheme yields a regular system with damping forces obtained from a linear combination of the Grunwald Letnikov coefficients with the past displacement terms. Thus, all the advantages of the Newmark numerical scheme can be utilized. In addition, statistical linearization is performed for random and deterministic excitations on non linear oscillators with dampers governed by fractional derivatives. 11
13 1.3 Organization of the Thesis This Thesis is organized as follows: Chapter 2 contains background information on the polyester mooring systems and on the use of fractional derivative models to model viscoelastic behavior and frequency dependent materials. Chapter 3 describes the proposed procedures to obtain a static and a dynamic stiffness using inservice real data. Pertinent mathematical background on the catenary differential equations is presented and numerical procedures such as the Gauss Newton non linear minimization algorithm to solve these equations are implemented. Chapter 4 describes how the proposed procedures can be used to avoid error contamination and presents the numerical results derived from the model. Chapter 5 presents the concept of oscillators with damping or stiffness forces governed by fractional derivatives of time and the mathematical background on fractional the derivatives in general is presented as a preliminary effort to capture better the dynamic behavior of polyester mooring lines. In addition it is shown that the advantages of the Grunwald Letnikov fractional derivative representation in conjunction with Newmark algorithm yield to an ordinary system with an updated stiffness modulus that is easier to solve. Finally numerical results are presented for sinusoid excitations. Chapter 6 presents the linearization technique to deal with non linear systems for deterministic and random excitation. Numerical results of linearized systems as well as comparison plots of these system's responses versus the numerical evaluation are presented. Chapter 7 contains the catenary equations for the fractional derivative constitutive model and the numerical results for the EA dynamic modulus. Chapter 8 contains the concluding remarks along with some plans for future work. 12
14 CHAPTER 2: Background on Mooring Lines An offshore structure, often referred as oil platform or oil rig is a large structure in the ocean used to house workers and machinery needed to drill wells in the sea bed, extract oil or natural gas and process it on board in order to refine it and ship it onshore using a channel of connecting pipes. Historically, the first form of offshore platform was constructed at the beginning of the WW II off the golf coast of Louisiana USA. Depending on the environmental circumstances and the conditions under which the platform will operate, the platform can be fixed in the sea bed with tall piles, standing on an artificial island or float. Most of the offshore platforms are located in the continental shelf but in the recent years due to the increased need of oil production, engineers pushed back the limits on the areas where platforms can operate. New technologies and highly advanced materials allowed drilling and exploration in ultra deep water environments, extreme weather conditions and operation in much larger distances from the shore. Fixed platforms are built on concrete or steel legs directly anchored on the sea bed, on top they provide the proper deck area for drilling rigs, production facilities and crew quarters. Due to their immobility these structures are designed for a long period of operation and are usually found in the Norwegian fjords and the Scottish firths. Unless there are special circumstances the operation of such platforms is economically viable up to 500 meters (around 1700 ft) beyond that depth the installation and the anchoring of the legs becomes excessively high. The compliant towers are platforms that consist of slender flexible towers and of a pile foundation supporting a conventional deck for drilling and production operations. They are on purposely designed to take large lateral deflections and can be operated in depths from 400 to 900 meters. One of the most common 13
15 applications is the Semi Submersible platforms. These platforms have hulls and are buoyant with the right amount of weight so that they can float in upright position without any external forces acting on to support them. They are practical due to their high mobility and due to the height they can remain quite balanced and operational even in extreme weather conditions. The anchoring is traditionally made with steel mooring lines and recently in 1997 for the first time in the world with polyester mooring systems in Brazil. The fact that polyester mooring systems are buoyant when submerged thus they are adding zero weight to the hull made the development of such structures appropriate for ultra deep water exploration up to 3 Km depths. There are 2 more major categories of offshore designs, the Tension Leg Platforms (TLP) and the Spars which are highly correlated as structures with the difference that the TLP's have the legs deployed in such a way that the vertical movement is eliminated. The performance of any mooring system is a function of the size and type of the moored vessel, the environmental forces acting, water depth and soil conditions of the sea bed. Under trying sea conditions the proper choice of anchors, clump weights, chains and cables becomes vital for keeping the vessel on site and for the mooring system survival. In designing a mooring system, one must define the vessel which needs to be kept in place and then apply proper mathematical models and analysis techniques to check out its adequacy and station keeping capability. Then the design needs to account for possible cable tensions along with possible failure modes, such as the breaking of cable in an extreme event or the dragging of the anchor. Cost benefit analysis is the most critical part and after that, the finalization of the design. 14
16 In 1997 the first polyester mooring line was introduced as a permanent mooring system in Brazil on a Semi Submersible offshore structure. Two other installations followed in 1998 and in 1999 by the same company operating in Brazil. Polyester mooring lines came about due to their superior strength and their light weight which is zeroed out when the ropes are submerged. Therefore, the use of polyester mooring systems results to smaller platforms and ground chains, smaller chain jacks and fairleads, which also produce cost saving beyond the impact on the hull. There are specific ways for the line to be installed and operate and every polyester mooring system has to have an Inspection, maintenance, repair and retirement plan (IMRR). Due to these requirements the line consists of many parts from the jack of the platform to the anchoring point in the sea bed. In order for the polyester to maintain the initial design mechanical properties, any contact to the sea bed must be avoided. In addition any moving particles that can be attached to the rope can obstruct its design operation and therefore there is a specific depth under which the first polyester rope can be installed in order to assure that the chances of living organisms being attached to the rope is minimal. Regulations also dictate the existence of some small segments in the line for sample testing after the installation and operation and therefore there have to be segments in which the insertion and removal can be performed quite easily. Consequently, these segments usually are installed in the upper parts of the lines. In most of the cases the line has a number of polyester rope segments, 2 chain segments one at each tip point and connectors in between. The last tip of the line ends up in a suction pile onto the sea bed and the first tip is jacked in the platform with proper wheels and jacks so the first polyester segment starts after a specific depth even if the platform is operating in extreme weather conditions. The jacks can adjust the length of the in and out board chain length so the platform can be relocated and fixed exactly above 15
17 the well. Analysis of multicomponent lines is a quite difficult procedure during the design. The fact that the currents and wind define the mean position of the platform poses challenges in the design teams. A two modulus design is most of the times the first approximation for the design. A static modulus that will be used to calculate the displacements of the platform for certain wind and wave loading and a Dynamic stiffness for the design of the spar in hostile environmental conditions. In a series of papers by Childres [14] the mooring system is considered from a more practical standpoint and the advantages of a multicomponent line over the single component lines is discussed. Niedzwecki and casadella[5] developed a numerical algorithm to solve the catenary equations for a line with cable and chain components. However, such lines aren't used in ultra deep water mooring systems where the depths are greater than usual. In addition lines have connectors which act like clumped weights and in this method they are not treated. Nath and Felix [6] considered a single point mooring system with a uniform cable and predict mooring line motion and the tensions resulting from wave forces. They also have implemented a numerical model limited though to certain depths and wave conditions. Wilson and Gabaccio [7] also considered a uniform cable and therefore their technique is limited to single steel catenaries from the bottom up to the fairlead. Tuah and Leonard [8] discussed a finite element model for predicting the dynamic viscoelastic response of a cable. The model is a three parameter linear visco elastic model proper for dynamic analysis of multicomponent systems. A powerful method to determine the geometry of the line and the static equilibrium forces is the Peyrot and Goilois [9] which is basically an algorithm that iteratively determines the final equilibrium point of the line. The initial guess is of great importance to the convergence of the solution. The 'quasistatic' cable analysis clearly deals with the dynamics of the anchoring system in a static 16
18 manner, whereby a static equilibrium state is assumed at each time step. This assumption is valid because the response of the moored vessel is normally outside the frequency range of the mooring system. However, in this kind of analysis the line dynamics are ignored and there are situations where the dynamics of the line are of great importance. Ansari and Khan [10] showed that dynamics of the line can be of higher significance than they thought they are by modeling each line component as a discrete dynamic system. However, the discussed line has a submerged weight whereas in the polyester line the only weights that are important are the clumped weights of the connectors and thus any dynamics of the line will primarily come from these weights. The loads that will determine the design phase are coming from the wind, currents and waves. Because of the stochastic nature of the wind, its properties vary with time and location. In standard meteorological practice, wind velocity and direction is predicted as an average over a given interval in time, varying from 1 minute to an hour [11]. Despite the fact that wind fluctuations are occurring around the mean value of the velocity and direction, these forces are considered small in comparison to hydrodynamic forces. Thus a steady wind velocity and direction is considered by most researchers enough to design [17,11,12]. It is natural that the constant winds in velocity and in direction will move the platform to a new equilibrium position and keep it there with small variations. Therefore the static stiffness becomes an essential tool in the design process because it leads to the calculation of the average displacement of the platform around the new equilibrium point. Currents on the other hand can not be treated in this fashion. Their complexity of occurrence is considered to be a result of several other combining phenomena. Primarily currents come from ocean circulation and create steady currents, from cyclic changes in lunar and solar gravities causing tidal currents and from wind and water density 17
19 differences. Because of their slow variation in the analysis one can model them as constant loads and carry on the design using the static stiffness coefficient. Currents also create other types of loading for the submerged parts of the platform besides the force from the impact. Friction of the parts with the water is also an important load along with the pressure drag. Wave loading is important for the design of the platform, primarily for the position of the platform above the well and secondarily for the operation of the sensitive machinery in the deck. There are many sea spectrum formulas available in the literature with significant differences due to their experimental origin [1316]. Differences also occur due to parameters used for the ordinates of the spectral curves. Compatible time histories can be obtained using these spectral representations and the analysis can be carried on for many different wave scenarios [39]. Interesting appears to be the fact that most of the design of offshore platforms assumes the wind, currents and wave loads to be collinear. However, data gathered from offshore structures operating in the gulf of Mexico indicated that during the passing of the Hurricane's eye from a specific location these loads are not collinear. Further research has been carried to provide a quantitative approach at the differences in the design assuming non collinear loading [18]. In the design process, first an uncoupled problem is solved for static and dynamic conditions using the estimates of the static and dynamic modulus. The uncoupled form of the problem involves the substitution of the line to a massless linear or non linear spring. Using this procedure the maximum line tension is obtained and therefore a coarse approximation of the line and jacks dimension can be obtained. Since this motion isn't a representative motion of the platform in real time operation, the question that needs to be answered is how this motion is correlated to the real motion of the platform, which 18
20 naturally corresponds to solving the coupled problem [22]. For the coupled problem the line is decomposed to many parts and the mass is attributed to each node connecting the line parts. The mass and the damping of the platform have to be accounted in the analysis. Before the actual platform is built a smaller model usually 1:60 is tested in the lab under various excitations. Results from the actual model and from the numerical algorithms are compared to help in the modeling process of the real platform. Fernands and Rossi [24] showed that modeling of polyester mooring lines can be accurately captured by a small diameter distorted polyester line in the actual 1:60 model. Furthermore, they recommend the modeling procedure to be conducted with the same material in order to capture the non linear characteristics of the polyester. In general, displacements of the coupled system and the uncoupled differ, pitch and surge in the quasi static analysis are not as reliable as the line tension and heave might be [19]. Another fundamental difficulty in the design process of FPS moored with polyester lines is the modulus of elasticity. Coupled or de coupled, linear or non linear analysis needs a modulus of elasticity to initiate the iterative procedures and a function to update this modulus with time or frequency. Several papers have been published on the mechanical properties of polyester [27], [23] [29] and the admitted conclusions is that the polyester rope stiffness increases with increasing mean load, decreases with increasing load range and decreases with increasing rate of loading. Fernandes Del Vecchio and Castro [21] showed that if one is to avoid the tension dependence, will not vary significantly from the true modulus. Fernandez Del Vecchio et. [21] proposed a model after a series of testing that provides the modulus of elasticity as a function of the mean load, the instantaneous tension and the load amplitude. Flory [30] provided an equivalent expression that provides the modulus as a function of the instantaneous tension, the dry density of the 19
21 rope and the breaking strength of the rope. Coupled analysis by Arcanda Tahar and M.H. Kim [20] showed that during an experiment, lines tend to have hardening and softening behavior as the strain increases and therefore a more sophisticated model than Bosnian's [29] is needed. As it has been indicated, the design process of a polyester mooring line is greatly different from the traditional steel line. The polyester rope stiffness is a function of many variables unlike the steel. For a spar FPS currents will usually control the design. Unlike, wind or wave which result in mean and dynamic loading on the structure, current loads tend to produce a large mean load on the hull that is aligned with the current flow plus transverse hull motions due to vortex induced vibrations. As a result the static stiffness of the polyester line becomes much more important than the dynamic stiffness [27]. It is shown that the effects of the mean loads on the modulus of elasticity are much greater than those of the dynamic loads [26]. To date, the industry has collected most of the data in dynamic stiffness and thus little is known about the static stiffness. Despite the complexity of defining the stiffness in the polyester rope, researchers have conducted long scale experiments with reasonable results. Casey and Banfield [27] conducted a large scale of experiments to provide more information in the measurements of the axial dynamic stiffness depending on many parameters. Concluding their research they suggest that the measurement of the axial dynamic stiffness to obtain values representative of the in service conditions is not a straight forward method. Due to the dynamic stiffness depending on the cycles of the loading, the stiffness measured from in service lines will highly depend on the time they have been operating. On the other hand, the static stiffness is highly correlated to initial pretension and mean load, and little knowledge exists about the actual in service values. Understanding the rope stiffness is important for 20
22 proper mooring and riser system design. The dynamic stiffness is different from the static stiffness. Mooring systems are usually controlled by the current conditions, the static stiffness should be known accurately for a successful design [27]. 21
23 CHAPTER 3: Modeling of the Polyester Lines and Static Solution 3.1 Introduction A specific offshore platform is considered herein having 11 polyester lines divided in 3 groups; during the installation of the polyester mooring system the exact points of the piles and the ground chains at the sea bed were identified using GPS. Therefore, all the necessary information about the position of the anchoring points in the sea bed is known. The platform is equipped with a GPS satellite system and accelerometers, thus making the position of the platform accurately known at each time point. Tension measurements are also available at the fairlead jacket points. Keeping into consideration that GPS systems are accurate below certain frequencies and the accelerometers are accurate for high frequencies, the accurate displacements can be obtained from the proper portion of the frequency spectrum. This technique is easy to implement and it is shown below. The accelerometer output is integrated in time twice and the fourier transform of the signal is obtained, then the low frequencies are discarded. Equivalently, the fourier transform is obtained for the GPS signal and the high frequencies are discarded. Fusing now in the frequency domain both fourier transforms and taking the inverse transform provides the accurate displacements. Having the six degrees of freedom of the platform in the time domain, a translation is performed to the fairlead points (Points where the chain is attached to the platform). Therefore, a displacement versus time history is available at each fairlead point. The components and their properties are known for each line. That means, each line has a chain part to be jacked in the platform, a connector that connects the chain to the first polyester segment and since no more than 1500 ft of continuous polyester rope can be manufactured connectors are connecting each polyester rope until 22
24 the line covers the distance to the bottom of the sea. However, since the polyester rope is sensitive to friction and particles that can be attached in its surface the final part of the line has to be chain in order to make sure that the polyester rope will not be contacting the sea bed. Therefore the overall line makeup has two chain segments polyester ropes in between and connectors. In order to understand better the properties of the polyester rope it is useful to understand the way this material is composed. Subrope yarn is constructed into 3strand subropes using a special machine, a braiding machine afterwards assembles 56 sub ropes with a filter cloth wrapped around the sub ropes prior to the jacket overbraid, forming the rope. Both ends of the rope are manually hand spliced by a pair of splicers. Sub ropes are spliced into a matched pair and thus each pair of sub ropes has four splices. The sub ropes are arranged in layers with eight sub ropes in a layer and seven layers. Between each layer a high modulus polyethylene cloth is installed around the sub ropes to provide added wear protection. The picture below shows a polyester rope and its components Petruska et all
25 Taking into account this manufacturing procedure of the sub ropes that assemble the rope, the first stretch that will be applied in the rope will produce an elastic and a plastic deformation. The elastic deformation will come from the elasticity of the sub ropes and the plastic one will come from the rearrangement of the twisted sub ropes inside the rope. The later is called the construction creep, and it is a form of instantaneous creep release. Aside from that the rope is subjected to regular creep, wear and fatigue. Unlike the certainty in the lengths of the connectors and chains, the lengths of the polyester segments involve uncertainty. Reports show that the lines were prestressed to fit the length demands during installation. Thus, a certain amount of 'construction creep' was released and the behavior of the polyester can be approximated as linear up to excitation loads exceeding the pretension level. Reports show that the pretension was at 40% of MBL (Maximum Breaking Load) and the maximum force measured at the fairleads at the hurricane peak event is approximately 30%, thus it is reasonable to assume a linear behavior of the polyester segments in every loading situation. However, the initial length after which this behavior can be assumed linear was estimated by assuming a static stiffness coefficient. Therefore, the initial zero stress length of each polyester segment is unknown. Recapitulating, tension measurements are available at the fairleads, each line makeup is fully determined except the length of the polyester segments which are prestressed to release the 'construction creep' and thus to behave linearly, the position of the spar and the anchoring points of each line at the sea bed are identified in a global axis system. 24
26 3.2 Modeling of the Polyester Mooring System Problem As it can be seen in Figures 3.1 and 3.2, each line comprises 15 components most of them connectors, chains, and 3 large polyester segments. In a global axis system the exact location of the fairlead can be obtained and the anchoring points are known; also the force at the fairlead is known. Having this information available along with the catenary equations, one can estimate the exact position of the end tip of each component. This is done by starting from a known position and tension of the initial point and knowing the properties of the component. Next iteratively proceeding from the first component (chain on the platform) to the last one (chain at the sea bed) the final point can be estimated and the shape of the submerged line can be determined. The shape of the line is estimated within certain error boundaries due to the fact that each part of the line must be calculated as a distinct catenary rope and the connectors are quite heavy short length parts that introduce errors as one proceeds from one component to the next. As the platform swings, the shape of the line changes according to the position and the force of the fairlead. More specifically the shape of the line changes when the inplane and vertical distance change, inplane distance is the horizontal distance between the anchoring point and the initial point of the line. Assuming that the zero stress length (L 0 ) is known by randomly choosing an initial angle <P, the coordinates of the final tip of the line can be calculated. Further, since the real anchoring points are known, there is an error vector R, shown in equation YjZtDepth R= 15 " (3D Y j X i InplaneX Displacement _»=i 25
27 that ultimately must be minimized with respect to some parameters which may be the angle <P and the EA modulus. In the same manner if the EA of the polyester components, is assumed known the angle and the L 0 can be the minimizing parameters. Assuming that certain algorithm minimizes the error R, convergence can be achieved for the minimizing parameters, L 0 and therefore for each time point these parameters can be identified. However, and EA are both unknown parameters along with the angle, and since a constant EA and L 0 can be assumed for calm sea conditions, these two parameters must remain constant in every calm sea excitation. Using the fact that in calm sea conditions the tension is fairly constant and the platform swings slightly the average position of the platform can be used for two different tension levels, the minimum calm sea tension, and the maximum calm sea tension. Thus an equivalent system of 2 unknowns with 2 equations can be set up where the unknowns are the EA and the L 0. Obviously, this system is a non linear system of equations and its solution will be obtained by numerical iteration. 26
28 Vertical Inplane Motion Figure 3.1. The linefromthe fairlead point of the platform to the anchoring point and the global and local coordinate systems Figure 3.2. A typical polyester mooring line in a inplane depiction. Known and unknown parameters of the line are also shown 27
29 3.2.1 Mathematical Background Considering Figure 3.3 the submerged infinitesimal rope section is under tension and water current loads, as well as its own submerged weight. Forming the equilibrium equation for the infinitesimal submerged rope segment in the X and Z directions, yields T+dTpgzApgdzA <p+dq> TpgzA Figure 3.3. The infinitesimal submerged rope section starting from the sea floor and dt  pgadz = [wsm(p  F(\ + TIEA)]ds (3.2) Td(p  pgazd(p = [wcoscp + D{\ + TI EA)]ds. (3.3) Thus subtracting the water pressure force from the tension and assuming F,D current forces equal to zero T' is defined. That is, and T = TpgzA, dt' = w sin (pds, T'dq) = wcosq>ds, (3.4) (3.5) (3.6) Further, combining equation 3.5 and equation 3.6 yields 28
30 Integrating the differential equation 3.7, gives dt \ _ sin <p T' cosp dip. (3.7) 1 ~ L o cos (p Decomposing the differential length in the two dimensions x and z yields (3.8) and dx = cos tpdp dz = sin ipdp. (3.9) (3.10) And correcting the initial length by the stretched length yields dp = ds(l + T/EA). (3.11) Using the sin and cosine of the angle <P at the infinitesimal triangle of Figure 3.3, and assuming that the angle is negligibly affected from the water pressure subtraction, yields dx T' = cos^(l + TIEA) * cos0>(l + riea) = coscp + ~^ds EA and = sm(p{\ + TIEA)nsm(p(\ + riea) = sm(p + s. ds EA (3.12) (3.13) Integrating equation 3.12 gives / rrr f COS^H EAcoscp 0 x  x Q = jcos^(l + TV EAcosq> 0 )ds. ]ds (3.14) (3.15) Combine 3.6 and 3.8 and substitute the cosine in equation 3.15 to change the integration variable.rvco. W jg (t+ _r_ )^ J wcosp ds EA cos <p 0 (3.16) Further introduce the variable <P~ and changing variables from ds to d(p yields, 29
31 _ f x x o ~ J V 2!% +?' )dq>, wcosq) EAcos<p 0 (3.17) and carrying out the multiplication yields '"ZLS^^+JLjS^SLrf,,. wcoscp EAw J cos <p (3.18) Carrying out the integration in equation 3.18 yields * = ^^%og w 1 cos#> 1 COS^9 0 + tan^? + tan^0. r n,2 cosc> n^ )+ ra yo (tan^tan^0) + x 0. (3.19) EAw Omitting from the present text the tedious calculations, one can follow the same procedure as above for Z direction and will finally derive for the two dimensions x and z representing the inplane horizontal displacement and the vertical displacement as shown in Figure 3.1. The equation in the Z direction yields, T 0 'cos(p 0 w 1 1 W 0 ' 2 cos> 0 (( COS^J COS^0 wea 2 cos (p 2 cos %)  tan <p 0 (tan <p  tan <p 0 + ) z n (3.20) The above stretched catenary equations provide the coordinates of the final point of a submerged rope if the coordinates of the initial tip are known. The coordinates of each fairlead are given by the procedure described above and has the following form. The GPS system is placed in an elevated known position on the platform and provides 'Northings' and 'Eastings'. Angular data are also provided by equipment on board (pitch, roll, yaw). 30
32 Initially from the GPS antenna one can calculate the Surge and Sway at the fairlead center linefromthe symbolic equation (GPS_Surge^ cos(yaw) sm(yaw) GPS _ Sway J L s'm(yaw) cos(yaw) Northings Eastings sm(pitch) sin(ro//) Height (3.21) In this equation the height is the relative height of the antenna with respect to the fairlead center line, Northings and Eastings are the GPS output, pitch, roll, yaw and surge, sway and heave are shown in figure 3.4. Figure 3.4. Displacements and rotations with respect to a certain axis system Accelerometers provide the surge sway and heave acceleration, and similarly can be integrated twice and be translated to the fairlead center line. Since Accelerometers and the GPS are fixed on the platform, the distortion of the platform itself with respect to the 31
33 platform movement is negligible, thus a rigid body transformation can be applied from the measuring points to the fairleads. The symbolic equation is, Surge Surge Sway S Sway Heave Heave dt 2 + cos(yaw) sin(yaw) 0 sin(yaw) cos(yaw) cos(pitch) sin(pitch) 0 sm(pitch) cos(pitch) cos(roll) 0 sin(ro//) sin(ro//) 0 cos(roll) (3.22) In this equation, the vector r is the relative position of the fairlead with respect to the fairlead centerline of the platform. Integration in time can be done in the frequency domain using multiplication. Then Fourier transform is performed and certain bands are kept from each transform's domain. That is, Surge Fj Sway Heave GPS_Surge = F GPS _ Sway GPS Heave Ace JSurge + F Ace_Sway  formo, foe) Ace Heave for_mfac,fsamp i ing l2) (3.23) In this equation F is the Fourier transform f ac is the given frequency up to which the GPS provides accurate measurements and beyond that value accelerometers are considered more accurate. The following equation provides the fused Fourier transform of the surge sway and heave and by taking the inverse Fourier yields 32
34 Surge Surge] Sway = F' F Sway >> (3.24) Heave Heave Fairlead displacements can be obtainedfromequation 3.22 using the appropriate R vector corresponding to each fairlead with respect to the fairlead centerline. 3.3 The GaussNewton nonlinear minimization algorithm Knowing the coordinates of the fairlead, the force at the line, the segments of the line and their properties, one can start with an initial guess for the angle between the chain and the platform and shoot down the line at the sea bed. Knowing the coordinates of the end tip, the error vector R of the horizontal distance and the vertical distance between the anchoring point end the tip of the line can be calculated, this vector now can be minimized using the Gauss Newton non linear minimization algorithm. Therefore, equation 3.1 becomes, z,,(l + TIEA)sm{S t )  Depth R = i (3.25) The output of the algorithm can be the right initial angle and the EA of each polyester segment or the angle and the L 0 of each polyester segment. Figure 3.5 below shows the iterations taking place to minimize the error (until the end tip of the line shoots down at the sea bed precisely at the anchoring point) 33
35 Figure 3.5. Iterative procedure of 'shooting down the line' until the error in equation 3.1 is minimized Numerical methods for minimization problems are well developed and quite common to use in engineering projects. However, their greatest weakness is associated with the initial guess, and the fact that they can only find a local minimum solution which most of the times isn't the global minimum. In addition, the problem of 'entrapment' is quite common and basically relates to the symptom of a local minimum being found but the algorithm is trapped inside it and stagnates for many iterations. In this case there are certain boundaries that need to be imposed. The stiffness modulus must be within certain range of values and the construction creep can't be greater than a large value of 10%. Specifically 0 (0, K12), EAE (min(ea),max(ea)) and Lo must satisfy the following relationshipl.l*lor>lo>lor where Lor is the original zero stress length installed prior to the construction creep removal and Lo is the assumed initial length with linear behavior after the construction creep has been removed. 34
36 Using the algorithm described with the above conditions a testing to calibrate the model was conducted with real and fictitious inputs and the errors were compared with the professional program PROFLEX that usesfiniteelements for the line. Knowing the initial value x x o= j, A. (326) the next step is determined by the 'backtracking' algorithm which makes sure that the equation f(x k +a k p k )<f(x k ) (3.27) holds for step k. Two conditions must hold and they are called the Armijo conditions. That is /(** + a kp k ) ^ /(** ) + <h a^fkpk ( 328 ) and Vf{x k + a kpk ) T p k > c 2 Vf T k p k, (3.29) where, equation 3.28 guarantees that the initial error is becoming smaller in every step, and equation 3.29 guarantees that the step towards the local minimum is large enough so that the algorithm does not 'stagnate', the symbol f denotes the quadratic object function that must be minimized and corresponds to the norm of the error vector R. Coefficients c x and c 2 have been adjusted for stability and fast convergence of the algorithm and depend on the sensitivity of the problem. The coefficient a usually is set to 1 but there are ways to be adjusted for faster convergence (Cauchy initial point). That is, 35
37 a = \ hi \\ J kp k \ (3.30) where J k represents the Jacobian matrix, and is formed numerically depending on the minimizing parameters and J = [drld6 dridt] (3.31) 1 m /(*)=^2>/(*). 1 7=1 (3.32) The Jacobian has been introduced in equation 3.31 and the gradient of the function f is given by the equation V/(*) = r y (x)vr y (x) = J(x) T R(x), (3.33) y=i where fj in this case is each component of the R vector columnwise, p vector is the step towards the local minimum and is given by equation T r\l / TT ; Pk=(j'jy(j'R). (3.34) Knowing the 'step' p the next x value can be obtained from the equation x k+ i=x k +a kpk (3.35) 36
38 CHAPTER 4: Numerical Results 4.1 Modeling connectors as catenaries Considering a line segment of length ds in Figure 4.1, force equilibrium demands that the change of the angle from <P to (f> + dq> provides an equal and opposite force to counterbalance the submerged weight. Note that if the dx curved segment is significantly heavy and becomes short (connectors are quite heavy and short) this angle becomes smaller and therefore the dt force increases to a quite large value in order for the equilibrium to hold. Thus, naturally a first attempt to lower the error will be to redistribute the weight of the connectors to a larger fictitious connector. The tables below show the comparison of the results from the custom made algorithm vis a vis commercial finite element package for inservice operational lines consisted of 3 connectors 2 polyester ropes and 2 chain segments. The accuracy of the custom algorithm in the tension and in the initial angle of the chain is of particular interest. Tables 4.1 through 4.3 show how the accuracy of the results improves by dealing with the 'effective' length of the connectors, where, the 'effective' length is the fictitious longer length for a lower distributed weight. Figure 4.1. Infinitesimal submerged rope segment under its submerged weight 37